Number of polynomials whose Galois group is a subgroup of the alternating group

Firstly, the conjecture whereof you speak (with an $\epsilon$ in the exponent) has been proved by yours truly (there is an arXiv.org preprint as of about six months ago).

Secondly, the most common "exceptional" situation is when the polynomial is reducible. It is clear that at least $O(B^{n-1})$ polynomials are reducible, and this is the truth, asymptotically, for $n>2.$

Thirdly, the Galois group is a subgroup of $A_n$ if and only if the discriminant is a perfect square. The obvious heuristic is that the probability that the value of a polynomial of degree $d$ is a perfect square is something like $1/B^{d/2}$ The degree of the discriminant is $2(n-1),$ which would indicate that alternating group is pretty thin on the ground.

ADDED LATER Experimental data (for the probability that a monic irreducible polynomial of degree $n$ and coefficients bounded by $B$ in absolute value has discriminant a perfect square) is consistent with the heuristic above when $n>3$ - the results are not clear for $n=3,$ and the question is vacuous for $n=2$ (a polynomial whose discriminant is a square is reducible).

The answer to this question really depends on how you count. The conjecture that "Galois groups containing transpositions make up a positive proportion of number fields and are the only groups that do" applies to counting fields by discriminant; when you count polynomials in a box instead, you get 100% S_n, as you say. So in the discriminant-counting setting, the alternating group is not in fact the second-most-common one.

If you want a lower bound for the number of alternating extensions, one way is just to construct a bunch of them out of S_n-extensions (Wikipedia) More geometrically; the reason we know the inverse Galois problem has a positive answer for A_n is that we can construct a parameterized family of polynomials whose Galois group is contained in A_n; then you get that there are B^a such polynomials with coefficients at most B, for some probably smallish rational number a.